Hi,
I came to this thread via google last night as I had a similar issue.
My ipad was stolen so I changed my log in email and password. (It was an old hotmail email not used in ages so not able to be reused either).
My new ipad air arrived so I instaled my backed up data from itunes and set about getting to grips with ios7.
I quickly found that FB wanted the password for my account, and ran into the same as above - settings, delete, won't delete, delete app, turn off ipad, reload app, and a pop up screen that wouldn't go away unless you pressed settings instead of cancel. etc etc,
Slept on it.
Today, I have got it sorted. I hope this will help others of you.
I deleted the FB app - hold finger on icon until jiggly icon and cross, then delete. then press close button to stop everything jiggling.
Plugged ipad into main computer and opened itunes, checked if I was on latest software etc, then clicked on apps in itunes and went down the list of apps - it shows every app you've ever had (unless you've deleted them from this list)
Here's where I think I beat ios7....I loaded fb app from the list of apps in my itunes rather than downloading it straight onto the ipad.
I set the ipad to sync as usual, then ejected it from the computer.
I opened the ipad, clicked on the icon for FB, and this time instead of it asking me for my password, I got the option to enter my email address and password. As I was doing this I got the usual pop up asking for my password, but this time when I pressed the cancel button, it didn't pop up again and again and again.
After what seemed like an age while the spinning circle of death loaded my FB page, I eventually got into FB, and noticed that the FB app I had loaded was an older version (possibly for iphone?) - this is what I think you need to get around ios7 - If your version of FB in apps is too new you may need to see if you have an older back up of your ipad/iphone in itunes that you can install to get around the cycle of the pop up.
On my ipad, I then opened the app store update and sure enough FB was waiting to be updated...clicked on the update and bingo! I'm now using FB with ios7 and my new email and password.
I hope this will work for at least one other person.